# Basic properties of numbers

BASIC PROPERTIES LIST:

There are two fundamental operations, addition and multiplication, that have the following basic properties, listed with their commonly designated names (*a,b,c *are real numbers):

MORE DETAILED VIEW OF THE BASIC AND SOME DERIVED PROPERTIES, DEFINITIONS AND EXAMPLES:

**Closure law:** The sum *a + b *and the product *a.b* or *ab* are unique real numbers; more specifically if *a* and *b* are positive (belong to R+), then *a + b *and *a . b* are positive

**Commutative law: ***a + b = b + a*: order does not matter in addition, *ab = ba*: order does not matter in multiplication.

**Associative law:** *a + (b + c) = (a+ b)+ c = a + b + c*: grouping does not matter in addition. *a(bc) = (ab)c = abc*: grouping does not matter in multiplication.

**Distributive law: ***a(b+ c) = ab + ac; (a+ b)c = ac + bc*: multiplication is distributive over addition.

**Identity law: **There is a number *0* with the property that *0 + a = a + 0 = a*; There is a number *1* with the property that *1.a = a .1 = a*

**Inverse law:** For any real number a there is a real number *-a* such that *a + (-a) = (-a) + a = 0 *For any nonzero real number *a* there is a real number *a*^{-1} such *that aa*^{-1}* = a*^{-1}*a = 1*; *-a* is called the **negative** of *a* ; *a*^{-1}* *is called the **reciprocal** of *a*

**example**: Associative and commutative Laws: Simplify (2 + x) + 7.

(2 + x) + 7 = (x + 2) + 7 Commutative Law

= x + (2 + 7) Associative Law

= x+9

**example** Show that (a+ b)(c+ d) = ac + ad + bc + bd.

(a+ b)(c + d) = a(c + d)+ b(c + d) Distributive law

= ac + ad + bc + bd Distributive Law

This rule is referred to as **FOIL** (First Outer Inner Last).

**Zero factor law:** For every real number a, a . 0 = 0 ; If a.b = 0, then either a = 0 or b = 0.

**Law for negatives: **-(-a) = a ; (-a)(-b) = ab ; -ab = (-a)b = a(-b) = -(-a)(-b) ; (-1)a = -a ;

**Definition of substraction:** a -b = a + (-b)

**Definition of division:** a / b = a . b^{-l}. therefore b^{-l} = 1 . b^{-l} = 1 / b ; Note that since the number 0 has no reciprocal, a / 0 is not defined.

**Properties of division: **-(a/b) = (-a)/b=a/(-b)= - (-a)/(-b) ; (-a)/(-b) = a/b ; a/b=c/d if and only if a.d = b.c ; a/b = (p.a)/(p.b) for any non-zero p

**Ordering properties:** The positive real numbers R+ have the following properties:

1. If *a* and *b* are in R+, then *a + b* and *ab* Are in R+

2. For every real number *a*, either *a* is in R+, or *a* is zero, or *-a* is in R+.

If *a* is in R+, a is called **positive**; if *-a* is in R+, a is called **negative**.

The number *a* **is less than** *b*, written *a < b*, if *b -a* is positive. Then b** is greater than** *a*,

written *b > a*. If *a* is either* ***less than or equal to** b, this is written a ≤ b. Then b is **greater**

**than or equal to*** a*, written *b ≥ a*.

**example: ***2 < 3* because *3 - 2 = 1* is positive. *-6 < 4* because *4 -(-6) = 10* is positive.

The following may be deduced from these definitions:

*a > 0* if and only if a is positive.

If *a ≠ 0*, then *a*^{2}*> 0*.

If *a<b*, then *a+c < b+c*.

If *a < b*, then *a.c > b.c* if *c < 0* and *a.c < bc* if *c > 0*

For any real number a, either *a > 0*, or *a = 0*, or *a < 0*.

If *a < b* and *b < c*, then *a < c*.

**The real number line: **Real numbers may be represented by points on a line in such a way that to each real number *a* there corresponds exactly one point on I, and vice versa:

**Absolute value of a number: **The absolute value of a, written lal, is defined as:

|*a*| = ( *a* if *a ≥ 0*; *-a* if *a < 0*)

**example: **|*10*| *= 10* ; |*-37.98*| *= 37.98*

**Properties of absolute value:**

|*a*| *≥ 0*

|*ab*|*=*|*a*| |*b*|

|*-a*|*=*|*a*|

|*a/b*|=|*a*|*/*|*b*|

|*a+b*|*≤*|*a*|*+*|*b*|

**Complex numbers: **The set C of numbers of the form a + bi, where a and b are real and i^{2} = -1, is called the complex numbers. Since every real number x can be written as x + 0.i, it follows that every real number is also a complex number.

**example: **3 + 2i, -5i are examples of nonreal complex numbers.

**Fractions properties:** The fraction properties follow from the basic properties and the definition of division. The most commonly used fraction properties are listed below:

**Order of operations:** In expressions involving multiple operations of different types, the fol-

lowing order is followed:

1. operations within grouping symbols are performed first. If grouping symbols are nested

inside other grouping symbols, operations are performed in order from the innermost outward.

2. exponents are performed before multiplications and divisions (unless rule 1 indicate otherwise)

3. perform multiplications and divisions from left to right, before

additions and subtractions, also from left to right) ( unless unless rule 1 and 2 indicate

otherwise).

The above rules for the order of operations are known under the abbreviation **PEMDAS** - Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction

**example:** Evaluate 10 - 8[(23 – 6)(3 + 4)]

10 - 8[(23 – 6)(3 + 4)] = 10 - 8[(17) . (7) ] = 10 – 8.[ 119]

= 10 - 952 = -942